Squares and Square Roots

Squares and Square Roots: An Introduction

After you are done with the preceding chapters, you will proceed to learning about squares and square roots in your CBSE Maths Class 8 Chapter 6 curriculum. This is one of the most interesting chapters that requires you to concentrate properly. 

Along with the various mathematical solutions, this chapter also helps a student understand the various properties of square roots. Additionally, you can learn various tips and tricks regarding the calculation of perfect squares and square roots.


What are Perfect Squares?

When you multiply any whole number with itself, you get a result, which is known as the perfect square of the first number.

Let us consider an example to clarify what a perfect square signifies. Suppose, the starting number as 12. When multiplied by itself or squared, the resulting figure is 144. You can say that 144 is the perfect square for the number 12.


Important Properties of all Square Numbers

The key to understanding squares and square roots is to first gauge a few common characteristics, which are shared by all square numbers. Learning them can help your calculations greatly – 

  • If a number has 3, 8, 2 or 7 in its units’ place, then it cannot be a perfect square number.

  • Numbers with 0, 4, 1, 6, 5 and 9 may or may not be perfect squares. For example, 36 has six in the units’ place and is the square of the number 6. Nevertheless, 136 also has six in the units’ place but is not a perfect square. Its square root is 11.66.


Square Root Calculation

When you know the square of a number but do not know the number itself, the procedure to discover this number is known as square root. It is basically the inverse function of squaring a figure.

Thus, now that you know the relationship between square and square roots, here is a few tips that should assist you in estimating square root from a given figure.


Useful Notes

You can easily calculate how many digits the square root of a given perfect square possesses by a simple formula.

Consider ‘x’ as the number of digits in the perfect square. Number of digits (n) in the square root is equal to x/2, where x is even. If x is odd, n = [x + 1]/2.

For example, let us consider the number 625. Here, x = 3, which is n odd number. Therefore, n = [3 + 1]/2 = 2. We can confirm this assertion as the square root of 625 is 25, which has two digits.


Estimating the Value of Square Roots

If you are in a hurry and need an approximate square root value, you can estimate the same using a simple technique. To demonstrate the same, let us take the number 671 as an example. It is not a perfect square. 

However, to estimate, let us consider the two perfect square numbers closest to 671, which are 625 and 676. Since 671 is closer to 676, whose square root is 26, you can assume that the square root of 676 is approximate 26 as well. 


Calculating Square Root Using Subtraction Method

In squares and square roots chapter, you will also learn about another method to derive square root from a perfect square. Suppose, the number as 25. The number of subsequent odd number subtractions you can undertake from the given number before reaching 0 should be its square root.

25 – 1 = 24

24 – 3 = 21

21 – 5 = 16

16 – 7 = 9

9 – 9 = 0

As you can see, there are 5 such subtractions before reaching 0. Therefore, we can say that the square root of 25 is 5. 


Multiple Choice Question

Q. Which of the following numbers will produce even squares?

  1. 981

  2. 783

  3. 268

  4. 547

Ans. (c) 268

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FAQ (Frequently Asked Questions)

1. How can you say at a Glance that the Numbers 1057 and 7928 are not Perfect Squares?

Ans. Only numbers ending with 0, 4, 5, 6, 1 and 9 can be perfect squares. Since both the numbers in the above question end with numbers other than these, we can say that the figures cannot be perfect squares. Numbers with 3, 8, 2 and 7 in the units’ place cannot be perfect squares. This is easier to understand with the help of examples.

For instance, 100, 144, 625, 36, 121 and 169 are examples of perfect squares, and each of them ends with the numbers mentioned above. However, apart from these numbers, if you have any others in the units place, you can straight away claim that those are not perfect squares.

2. What is the Long Division Method of Determining Square Root?

Ans. Long division method is the most accurate way to determine the square root for any given number. It can be a time-consuming process and would require the use of pen and paper to conduct such calculations. However, this is the most common way for square root determination, when you require an exact figure and not an estimate.

3. What Does the Sum of Consecutive Natural Numbers Mean?

Ans. The rule claims that every square includes the summation of two consecutive numbers. Let us take for an example the number 25, whose square root (n) is 5. The first number is therefore (n² + 1)/2, while the second number is (n² – 1)/2. Here, the first number will be 24/2, which is equal to 12, while the second number is 26/2 that equals 13. The summation of 12 and 13 gives us the square number, which is 25.

Let us take another example to highlight this point. For example, 81 is the square of 9. Thus n = 9. Here, the first number would be (9² + 1)/2 and the second number would be (9² – 1)/2. Thus, it would be 82/2 + 80/2. Therefore, we once again attain the square number 81 from this method.

4. Can one Number have more than one Square Root?

Ans. Every number has two square root values. The first one is the positive value, also known as the principal root. The second value is a negative root. For example, the square root for 16 would be 4 and -4. Similarly, the two roots for 144 would be 12 and -12.